Covariance Matrices, Portfolio Variance, and Diversification

/**
 * Fintech Math Bootcamp · Lesson 110 of 120
 * Covariance Matrices, Portfolio Variance, and Diversification
 * Module 11: Financial Risk and Performance Statistics
 *
 * Scenario: Why two individually quiet holdings can become a risky combination
 * Rule:     portfolio variance = wᵀΣw
 *
 * Try it:   What changed between the two portfolio calculations?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/covariance-matrices-portfolio-variance-and-diversification/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson110() {
  const w=[.5,.5],vol=[.01,.01];
  const variance=(rho:number)=>{ // w' Σ w
    const cov=[[vol[0]**2,rho*vol[0]*vol[1]],[rho*vol[0]*vol[1],vol[1]**2]];
    return w.reduce((s,wi,i)=>s+wi*w.reduce((t,wj,j)=>t+cov[i][j]*wj,0),0);};
  const result={atMinusHalf:variance(-.5),atPointNine:variance(.9),
    dollarSDLow:100000*Math.sqrt(variance(-.5)),
    dollarSDHigh:100000*Math.sqrt(variance(.9))};
  return result;
}

export const checkedResult = {"atMinusHalf":0.000025,"atPointNine":0.000095,"dollarSDLow":500,"dollarSDHigh":974.6794344808964};

// Run this file directly: npx tsx lessons/11-financial-risk-and-performance-statistics/110-covariance-matrices-portfolio-variance-and-diversification.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson110(), null, 2));
}

# Fintech Math Bootcamp · Lesson 110 of 120
# Covariance Matrices, Portfolio Variance, and Diversification
# Module 11: Financial Risk and Performance Statistics
#
# Scenario: Why two individually quiet holdings can become a risky combination
# Rule:     portfolio variance = wᵀΣw
#
# Try it:   What changed between the two portfolio calculations?
#
# Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/covariance-matrices-portfolio-variance-and-diversification/
# Free course:    https://courses.thefintechbuilder.com
# Synthetic teaching example, not financial advice or a production library.
#
# Run: python main.py

import json
import math


def lesson_110():
    w, vol = [0.5, 0.5], [0.01, 0.01]

    def variance(rho):  # w' Sigma w
        cov = [[vol[0] ** 2, rho * vol[0] * vol[1]], [rho * vol[0] * vol[1], vol[1] ** 2]]
        total = 0.0
        for i, wi in enumerate(w):
            inner = 0.0
            for j, wj in enumerate(w):
                inner += cov[i][j] * wj
            total += wi * inner
        return total

    return {
        "atMinusHalf": variance(-0.5),
        "atPointNine": variance(0.9),
        "dollarSDLow": 100000 * math.sqrt(variance(-0.5)),
        "dollarSDHigh": 100000 * math.sqrt(variance(0.9)),
    }


if __name__ == "__main__":
    print(json.dumps(lesson_110(), indent=2))

/*
 * Fintech Math Bootcamp - Lesson 110 of 120
 * Covariance Matrices, Portfolio Variance, and Diversification
 * Module 11: Financial Risk and Performance Statistics
 *
 * Scenario: Why two individually quiet holdings can become a risky combination
 * Rule:     portfolio variance = w^T Sigma w
 *
 * Try it:   What changed between the two portfolio calculations?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/covariance-matrices-portfolio-variance-and-diversification/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 *
 * Run: javac Main.java && java Main
 */

import java.util.*;

public class Main {
    static final double[] W = {.5, .5};
    static final double[] VOL = {.01, .01};

    // w' Sigma w
    static double variance(double rho) {
        double[][] cov = {
            {Math.pow(VOL[0], 2), rho * VOL[0] * VOL[1]},
            {rho * VOL[0] * VOL[1], Math.pow(VOL[1], 2)}
        };
        double total = 0;
        for (int i = 0; i < W.length; i++) {
            double inner = 0;
            for (int j = 0; j < W.length; j++) inner += cov[i][j] * W[j];
            total += W[i] * inner;
        }
        return total;
    }

    static Map<String, Object> lesson110() {
        return obj("atMinusHalf", variance(-.5), "atPointNine", variance(.9),
                "dollarSDLow", 100000 * Math.sqrt(variance(-.5)),
                "dollarSDHigh", 100000 * Math.sqrt(variance(.9)));
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson110(), ""));
    }

    // Minimal JSON writer: insertion-ordered Map, List, Number, Boolean, String and null.
    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof String) return quote((String) value);
        if (value instanceof Boolean) return value.toString();
        if (value instanceof Double) return formatNumber((Double) value);
        if (value instanceof Number) return value.toString();
        String inner = indent + "  ";
        StringBuilder sb = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            sb.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> e : map.entrySet()) {
                sb.append(inner).append(quote(e.getKey().toString())).append(": ").append(toJson(e.getValue(), inner));
                sb.append(++i < map.size() ? ",\n" : "\n");
            }
            return sb.append(indent).append("}").toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        sb.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            sb.append(inner).append(toJson(list.get(i), inner)).append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return sb.append(indent).append("]").toString();
    }

    // Integral doubles print without ".0", as JavaScript does; others use Java's round-trip form.
    static String formatNumber(double x) {
        if (x == Math.rint(x) && Math.abs(x) < 1e15) return Long.toString((long) x);
        return Double.toString(x);
    }

    static String quote(String s) {
        StringBuilder sb = new StringBuilder("\"");
        for (char c : s.toCharArray()) {
            if (c == '"' || c == '\\') sb.append('\\').append(c);
            else if (c < 0x20) sb.append(String.format("\\u%04x", (int) c));
            else sb.append(c);
        }
        return sb.append('"').toString();
    }

    // Builds an insertion-ordered object from alternating keys and values.
    static Map<String, Object> obj(Object... keysAndValues) {
        Map<String, Object> map = new LinkedHashMap<String, Object>();
        for (int i = 0; i < keysAndValues.length; i += 2) map.put((String) keysAndValues[i], keysAndValues[i + 1]);
        return map;
    }

    static List<Object> list(double... values) {
        List<Object> out = new ArrayList<Object>();
        for (double v : values) out.add(v);
        return out;
    }
}

// Fintech Math Bootcamp · Lesson 110 of 120
// Covariance Matrices, Portfolio Variance, and Diversification
// Module 11: Financial Risk and Performance Statistics
//
// Scenario: Why two individually quiet holdings can become a risky combination
// Rule:     portfolio variance = wᵀΣw
//
// Try it:   What changed between the two portfolio calculations?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/covariance-matrices-portfolio-variance-and-diversification/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: go run main.go

package main

import (
	"encoding/json"
	"fmt"
	"math"
)

type Result struct {
	AtMinusHalf  float64 `json:"atMinusHalf"`
	AtPointNine  float64 `json:"atPointNine"`
	DollarSDLow  float64 `json:"dollarSDLow"`
	DollarSDHigh float64 `json:"dollarSDHigh"`
}

func lesson110() Result {
	w, vol := []float64{.5, .5}, []float64{.01, .01}
	variance := func(rho float64) float64 { // w' Sigma w
		cov := [2][2]float64{
			{math.Pow(vol[0], 2), rho * vol[0] * vol[1]},
			{rho * vol[0] * vol[1], math.Pow(vol[1], 2)},
		}
		total := 0.0
		for i, wi := range w {
			inner := 0.0
			for j, wj := range w {
				inner += cov[i][j] * wj
			}
			total += wi * inner
		}
		return total
	}
	return Result{
		AtMinusHalf:  variance(-.5),
		AtPointNine:  variance(.9),
		DollarSDLow:  100000 * math.Sqrt(variance(-.5)),
		DollarSDHigh: 100000 * math.Sqrt(variance(.9)),
	}
}

func main() {
	out, err := json.MarshalIndent(lesson110(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 110 of 120
// Covariance Matrices, Portfolio Variance, and Diversification
// Module 11: Financial Risk and Performance Statistics
//
// Scenario: Why two individually quiet holdings can become a risky combination
// Rule:     portfolio variance = wᵀΣw
//
// Try it:   What changed between the two portfolio calculations?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/covariance-matrices-portfolio-variance-and-diversification/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: g++ -std=c++17 -O1 -o lesson main.cpp && ./lesson

#include <algorithm>
#include <charconv>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <deque>
#include <iostream>
#include <limits>
#include <map>
#include <optional>
#include <stdexcept>
#include <string>
#include <utility>
#include <variant>
#include <vector>

// Minimal JSON value: objects keep insertion order; numbers print in shortest round-trip form.
struct Json {
    enum class Kind { Null, Bool, Number, String, Array, Object };
    Kind kind = Kind::Null;
    bool flag = false;
    double number = 0.0;
    std::string text;
    std::vector<std::string> keys;  // object keys, parallel to items
    std::vector<Json> items;        // array items, or object values
};

Json jnull() { return Json{}; }
Json jbool(bool value) { Json j; j.kind = Json::Kind::Bool; j.flag = value; return j; }
Json jnum(double value) { Json j; j.kind = Json::Kind::Number; j.number = value; return j; }
Json jstr(const std::string& value) { Json j; j.kind = Json::Kind::String; j.text = value; return j; }
Json jarr(std::vector<Json> items) { Json j; j.kind = Json::Kind::Array; j.items = std::move(items); return j; }

Json jobj(std::vector<std::pair<std::string, Json>> fields) {
    Json j;
    j.kind = Json::Kind::Object;
    for (auto& field : fields) {
        j.keys.push_back(field.first);
        j.items.push_back(std::move(field.second));
    }
    return j;
}

Json jnums(const std::vector<double>& values) {
    std::vector<Json> items;
    for (double v : values) items.push_back(jnum(v));
    return jarr(std::move(items));
}

Json jnums(const std::vector<std::optional<double>>& values) {
    std::vector<Json> items;
    for (const auto& v : values) items.push_back(v ? jnum(*v) : jnull());
    return jarr(std::move(items));
}

std::string formatNumber(double value) {
    char buffer[64];
    auto result = std::to_chars(buffer, buffer + sizeof buffer, value);  // shortest round-trip form
    return std::string(buffer, result.ptr);
}

std::string quote(const std::string& text) {
    std::string out = "\"";
    for (char c : text) {
        if (c == '"' || c == '\\') {
            out += '\\';
            out += c;
        } else if (static_cast<unsigned char>(c) < 0x20) {
            char buffer[8];
            std::snprintf(buffer, sizeof buffer, "\\u%04x", static_cast<unsigned>(c));
            out += buffer;
        } else {
            out += c;
        }
    }
    return out + "\"";
}

void writeJson(std::ostream& out, const Json& value, const std::string& indent = "") {
    const std::string inner = indent + "  ";
    switch (value.kind) {
        case Json::Kind::Null: out << "null"; return;
        case Json::Kind::Bool: out << (value.flag ? "true" : "false"); return;
        case Json::Kind::Number: out << formatNumber(value.number); return;
        case Json::Kind::String: out << quote(value.text); return;
        case Json::Kind::Array:
        case Json::Kind::Object: {
            const bool isObject = value.kind == Json::Kind::Object;
            if (value.items.empty()) {
                out << (isObject ? "{}" : "[]");
                return;
            }
            out << (isObject ? "{\n" : "[\n");
            for (std::size_t i = 0; i < value.items.size(); ++i) {
                out << inner;
                if (isObject) out << quote(value.keys[i]) << ": ";
                writeJson(out, value.items[i], inner);
                out << (i + 1 < value.items.size() ? ",\n" : "\n");
            }
            out << indent << (isObject ? "}" : "]");
        }
    }
}

Json lesson110() {
    const std::vector<double> w{.5, .5}, vol{.01, .01};
    auto variance = [&](double rho) {  // w' Sigma w
        const double cov[2][2] = {{std::pow(vol[0], 2), rho * vol[0] * vol[1]},
                                  {rho * vol[0] * vol[1], std::pow(vol[1], 2)}};
        double total = 0;
        for (std::size_t i = 0; i < w.size(); ++i) {
            double inner = 0;
            for (std::size_t j = 0; j < w.size(); ++j) inner += cov[i][j] * w[j];
            total += w[i] * inner;
        }
        return total;
    };
    return jobj({{"atMinusHalf", jnum(variance(-.5))},
                 {"atPointNine", jnum(variance(.9))},
                 {"dollarSDLow", jnum(100000 * std::sqrt(variance(-.5)))},
                 {"dollarSDHigh", jnum(100000 * std::sqrt(variance(.9)))}});
}

int main() {
    writeJson(std::cout, lesson110());
    std::cout << "\n";
    return 0;
}

// Fintech Math Bootcamp · Lesson 110 of 120
// Covariance Matrices, Portfolio Variance, and Diversification
// Module 11: Financial Risk and Performance Statistics
//
// Scenario: Why two individually quiet holdings can become a risky combination
// Rule:     portfolio variance = wᵀΣw
//
// Try it:   What changed between the two portfolio calculations?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/covariance-matrices-portfolio-variance-and-diversification/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: rustc -O main.rs && ./main

#![allow(dead_code)]

/// Minimal JSON value; objects keep insertion order.
enum Json {
    Null,
    Bool(bool),
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, Json)>),
}

fn obj(fields: Vec<(&str, Json)>) -> Json {
    Json::Obj(fields.into_iter().map(|(k, v)| (k.to_string(), v)).collect())
}

fn nums(values: &[f64]) -> Json {
    Json::Arr(values.iter().map(|&v| Json::Num(v)).collect())
}

fn optional_nums(values: &[Option<f64>]) -> Json {
    Json::Arr(values.iter().map(|v| v.map_or(Json::Null, Json::Num)).collect())
}

fn quote(text: &str) -> String {
    let mut out = String::from("\"");
    for c in text.chars() {
        match c {
            '"' => out.push_str("\\\""),
            '\\' => out.push_str("\\\\"),
            c if (c as u32) < 0x20 => out.push_str(&format!("\\u{:04x}", c as u32)),
            c => out.push(c),
        }
    }
    out.push('"');
    out
}

// Display for f64 prints the shortest string that round-trips, as JavaScript does.
fn write_json(value: &Json, indent: &str, out: &mut String) {
    let inner = format!("{}  ", indent);
    match value {
        Json::Null => out.push_str("null"),
        Json::Bool(b) => out.push_str(if *b { "true" } else { "false" }),
        Json::Num(n) => out.push_str(&format!("{}", n)),
        Json::Str(s) => out.push_str(&quote(s)),
        Json::Arr(items) if items.is_empty() => out.push_str("[]"),
        Json::Obj(fields) if fields.is_empty() => out.push_str("{}"),
        Json::Arr(items) => {
            out.push_str("[\n");
            for (i, item) in items.iter().enumerate() {
                out.push_str(&inner);
                write_json(item, &inner, out);
                out.push_str(if i + 1 < items.len() { ",\n" } else { "\n" });
            }
            out.push_str(indent);
            out.push(']');
        }
        Json::Obj(fields) => {
            out.push_str("{\n");
            for (i, (key, item)) in fields.iter().enumerate() {
                out.push_str(&inner);
                out.push_str(&quote(key));
                out.push_str(": ");
                write_json(item, &inner, out);
                out.push_str(if i + 1 < fields.len() { ",\n" } else { "\n" });
            }
            out.push_str(indent);
            out.push('}');
        }
    }
}

fn lesson_110() -> Json {
    let (w, vol): ([f64; 2], [f64; 2]) = ([0.5, 0.5], [0.01, 0.01]);
    // w' Sigma w
    let variance = |rho: f64| -> f64 {
        let cov = [
            [vol[0].powi(2), rho * vol[0] * vol[1]],
            [rho * vol[0] * vol[1], vol[1].powi(2)],
        ];
        w.iter().enumerate().fold(0.0, |s, (i, wi)| {
            s + wi * w.iter().enumerate().fold(0.0, |t, (j, wj)| t + cov[i][j] * wj)
        })
    };
    obj(vec![
        ("atMinusHalf", Json::Num(variance(-0.5))),
        ("atPointNine", Json::Num(variance(0.9))),
        ("dollarSDLow", Json::Num(100000.0 * variance(-0.5).sqrt())),
        ("dollarSDHigh", Json::Num(100000.0 * variance(0.9).sqrt())),
    ])
}

fn main() {
    let mut out = String::new();
    write_json(&lesson_110(), "", &mut out);
    println!("{}", out);
}

// Fintech Math Bootcamp · Lesson 110 of 120
// Covariance Matrices, Portfolio Variance, and Diversification
// Module 11: Financial Risk and Performance Statistics
//
// Scenario: Why two individually quiet holdings can become a risky combination
// Rule:     portfolio variance = wᵀΣw
//
// Try it:   What changed between the two portfolio calculations?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/covariance-matrices-portfolio-variance-and-diversification/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: dotnet run

using System;
using System.Collections.Generic;
using System.Globalization;
using System.Linq;
using System.Text.Encodings.Web;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true, Encoder = JavaScriptEncoder.UnsafeRelaxedJsonEscaping };
Console.WriteLine(JsonSerializer.Serialize(Lesson110(), options));

static object Lesson110()
{
    double[] w = { .5, .5 }, vol = { .01, .01 };
    double Variance(double rho) // w' Sigma w
    {
        double[,] cov =
        {
            { Math.Pow(vol[0], 2), rho * vol[0] * vol[1] },
            { rho * vol[0] * vol[1], Math.Pow(vol[1], 2) },
        };
        double total = 0;
        for (int i = 0; i < w.Length; i++)
        {
            double inner = 0;
            for (int j = 0; j < w.Length; j++) inner += cov[i, j] * w[j];
            total += w[i] * inner;
        }
        return total;
    }
    return new
    {
        atMinusHalf = Variance(-.5),
        atPointNine = Variance(.9),
        dollarSDLow = 100000 * Math.Sqrt(Variance(-.5)),
        dollarSDHigh = 100000 * Math.Sqrt(Variance(.9)),
    };
}